At any \(K\) ,
in the manner fully consistent with Stone theorem [1]},
Hamiltonians \(H=Z_0\,\)  have to be self-adjoint
in
\({\cal H}_{phys}= {\cal R}_0\) .
Simultaneously,
all  of the relevant mathematical constructions
have to be performed
in working space
\({\cal R}_{N_{}}={\cal H}_{math}\) . Contextually,
the same Hamiltonian
can be called
Hermitian or non-Hermitian.
In the Hermitian case one simply
returns to the conventional probabilistic
interpretation of the unitary system in question.
In this sense
the physical interpretation of the predictions of the GSP
theory using \(K>1\) 
remains unchanged.
Only our mathematical working space
ceases to be equivalent to the correct
physical Hilbert space
so that the
operators \(\Lambda \)  representing the observables
are defined, in
\({\cal R}_{N_{}}={\cal H}_{math}\) , as non-Hermitian.
